The quick answer is the Cold War. The context surrounding the assassination of Ngo Dinh Diem provides the long answer.
After WWII the US and the USSR were engaged in the Cold War. The time period from 1949-1963 saw China, North Korea, and Cuba become outright communist countries, and a short lived regime sympathetic to Moscow appeared in Guatemala. This created great fear in the US foreign policy apparatus that the Cold War was being lost one proxy state at a time. This fear is a large contributing factor to why the US had a strong commitment to Ngo Dinh Diem.
In 1963, Ngo Dinh Diem was viewed by many in the US foreign policy apparatus as having lost control of the population, and unable to defeat the communists he was at war with. The famous image of the Buddhist monk lighting himself on fire had just been published in the New York Times, and Kennedy's advisors lost faith in Ngo Dinh Diem.
In this context (fear of the spread of communism) Kennedy gave implicit approval of the assassination operation. According to Stephen Kinzer's book Overthrow Kennedy never gave explicit approval, and regretted the death of Ngo Dinh Diem.
For further reading:
For the Common Defense @ pp. 573-576
Overthrow @ pp. 149, 152-169